Mixtilinear incircles of a triangle

In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex ${\displaystyle A}$ is called the ${\displaystyle A}$-mixtilinear incircle. Every triangle has three unique mixtilinear incircles, one corresponding to each vertex.

${\displaystyle A}$-Mixtilinear incircle of triangle ${\displaystyle ABC}$

Proof of existence and uniqueness

The ${\displaystyle A}$-excircle of triangle ${\displaystyle ABC}$ is unique. Let ${\displaystyle \Phi }$ be a transformation defined by the composition of an inversion centered at ${\displaystyle A}$ with radius ${\displaystyle {\sqrt {AB\cdot AC}}}$ and a reflection with respect to the angle bisector on ${\displaystyle A}$. Since inversion and reflection are bijective and preserve touching points, then ${\displaystyle \Phi }$ does as well. Then, the image of the ${\displaystyle A}$-excircle under ${\displaystyle \Phi }$ is a circle internally tangent to sides ${\displaystyle AB,AC}$ and the circumcircle of ${\displaystyle ABC}$, that is, the ${\displaystyle A}$-mixtilinear incircle. Therefore, the ${\displaystyle A}$-mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to ${\displaystyle B}$ and ${\displaystyle C}$. ^[1]

Construction

The hexagon ${\displaystyle XCABYT_{A}}$ and the intersections ${\displaystyle D,I,E}$ of its 3 pairs of opposite sides.

The ${\displaystyle A}$-mixtilinear incircle can be constructed with the following sequence of steps. ^[2]

1. Draw the incenter ${\displaystyle I}$ by intersecting angle bisectors.
2. Draw a line through ${\displaystyle I}$ perpendicular to the line ${\displaystyle AI}$, touching lines ${\displaystyle AB}$ and ${\displaystyle AC}$ at points ${\displaystyle D}$ and ${\displaystyle E}$ respectively. These are the tangent points of the mixtilinear circle.
3. Draw perpendiculars to ${\displaystyle AB}$ and ${\displaystyle AC}$ through points ${\displaystyle D}$ and ${\displaystyle E}$ respectively and intersect them in ${\displaystyle O_{A}}$. ${\displaystyle O_{A}}$ is the center of the circle, so a circle with center ${\displaystyle O_{A}}$ and radius ${\displaystyle O_{A}E}$ is the mixtilinear incircle

This construction is possible because of the following fact:

Lemma

The incenter is the midpoint of the touching points of the mixtilinear incircle with the two sides.

Proof

Let ${\displaystyle \Gamma }$ be the circumcircle of triangle ${\displaystyle ABC}$ and ${\displaystyle T_{A}}$ be the tangency point of the ${\displaystyle A}$-mixtilinear incircle ${\displaystyle \Omega _{A}}$ and ${\displaystyle \Gamma }$. Let ${\displaystyle X\neq T_{A}}$ be the intersection of line ${\displaystyle T_{A}D}$ with ${\displaystyle \Gamma }$ and ${\displaystyle Y\neq T_{A}}$ be the intersection of line ${\displaystyle T_{A}E}$ with ${\displaystyle \Gamma }$. Homothety with center on ${\displaystyle T_{A}}$ between ${\displaystyle \Omega _{A}}$ and ${\displaystyle \Gamma }$ implies that ${\displaystyle X,Y}$ are the midpoints of ${\displaystyle \Gamma }$ arcs ${\displaystyle AB}$ and ${\displaystyle AC}$ respectively. The inscribed angle theorem implies that ${\displaystyle X,I,C}$ and ${\displaystyle Y,I,B}$ are triples of collinear points. Pascal's theorem on hexagon ${\displaystyle XCABYT_{A}}$ inscribed in ${\displaystyle \Gamma }$ implies that ${\displaystyle D,I,E}$ are collinear. Since the angles ${\displaystyle \angle {DAI}}$ and ${\displaystyle \angle {IAE}}$ are equal, it follows that ${\displaystyle I}$ is the midpoint of segment ${\displaystyle DE}$. ^[1]

Other properties

Radius

The following formula relates the radius ${\displaystyle r}$ of the incircle and the radius ${\displaystyle \rho _{A}}$ of the ${\displaystyle A}$-mixtilinear incircle of a triangle ${\displaystyle ABC}$:$${\displaystyle r=\rho _{A}\cdot \cos ^{2}{\frac {\alpha }{2}}}$$

where ${\displaystyle \alpha }$ is the magnitude of the angle at ${\displaystyle A}$. ^[3]

Relationship with points on the circumcircle

• The midpoint of the arc ${\displaystyle BC}$ that contains point ${\displaystyle A}$ is on the line ${\displaystyle T_{A}I}$. ^{[4] [5]}
• The quadrilateral ${\displaystyle T_{A}XAY}$ is harmonic, which means that ${\displaystyle T_{A}A}$ is a symmedian on triangle ${\displaystyle XT_{A}Y}$. ^[1]

Circles related to the tangency point with the circumcircle

${\displaystyle T_{A}BDI}$ and ${\displaystyle T_{A}CEI}$ are cyclic quadrilaterals. ^[4]

Spiral similarities

${\displaystyle T_{A}}$ is the center of a spiral similarity that maps ${\displaystyle B,I}$ to ${\displaystyle I,C}$ respectively. ^[1]

Relationship between the three mixtilinear incircles

Lines joining vertices and mixtilinear tangency points

The three lines joining a vertex to the point of contact of the circumcircle with the corresponding mixtilinear incircle meet at the external center of similitude of the incircle and circumcircle. ^[3] The Online Encyclopedia of Triangle Centers lists this point as X(56). ^[6] It is defined by trilinear coordinates: $${\displaystyle {\frac {a}{c+a-b}}:{\frac {b}{c+a-b}}:{\frac {c}{a+b-c}},}$$ and barycentric coordinates: $${\displaystyle {\frac {a^{2}}{b+c-a}}:{\frac {b^{2}}{c+a-b}}:{\frac {c^{2}}{a+b-c}}.}$$

Radical center

The radical center of the three mixtilinear incircles is the point ${\displaystyle J}$ which divides ${\displaystyle OI}$ in the ratio: $${\displaystyle OJ:JI=2R:-r}$$where ${\displaystyle I,r,O,R}$ are the incenter, inradius, circumcenter and circumradius respectively. ^[5]

References

1. Baca, Jafet. "On Mixtilinear Incircles" (PDF). Retrieved October 27, 2021.
2. Weisstein, Eric W. "Mixtilinear Incircles". mathworld.wolfram.com. Retrieved 2021-10-31.
3. Yui, Paul (April 23, 2018). "Mixtilinear Incircles". The American Mathematical Monthly. 106 (10): 952–955. doi:10.1080/00029890.1999.12005146 . Retrieved October 27, 2021.
4. Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States of America: MAA. p. 68. ISBN   978-1-61444-411-4.
5. Nguyen, Khoa Lu (2006). "On Mixtilinear Incircles and Excircles" (PDF). Retrieved November 27, 2021.
6. "ENCYCLOPEDIA OF TRIANGLE CENTERS". faculty.evansville.edu. Retrieved 2021-10-31.